All ACT Math Resources
Example Question #9 : 45/45/90 Right Isosceles Triangles
What is the perimeter of an isosceles right triangle with an hypotenuse of length ?
Your right triangle is a triangle. It thus looks like this:
Now, you know that you also have a reference triangle for triangles. This is:
This means that you can set up a ratio to find . It would be:
Your triangle thus could be drawn like this:
Now, notice that you can rationalize the denominator of :
Thus, the perimeter of your figure is:
Example Question #3 : Isosceles Triangles
What is the perimeter of an isosceles right triangle with an area of ?
Recall that an isosceles right triangle is also a triangle. Your reference figure for such a shape is:
Now, you know that the area of a triangle is:
For this triangle, though, the base and height are the same. So it is:
Now, we have to be careful, given that our area contains . Let's use , for "side length":
Thus, . Now based on the reference figure above, you can easily see that your triangle is:
Therefore, your perimeter is:
Example Question #11 : 45/45/90 Right Isosceles Triangles
A tree is feet tall and is planted in the center of a circular bed with a radius of feet. If you want to stabalize the tree with ropes going from its midpoint to the border of the bed, how long will each rope measure?
This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem giving us . If then .
Example Question #12 : 45/45/90 Right Isosceles Triangles
An isosceles right triangle has a hypotenuse of length . What is the perimeter of this triangle, in terms of ?
The ratio of sides to hypotenuse of an isosceles right triangle is always . With this in mind, setting as our hypotenuse means we must have leg lengths equal to:
Since the perimeter has two of these legs, we just need to multiply this by and add the result to our hypothesis:
So, our perimeter in terms of is: